Bogdanov–Takens and Triple Zero Bifurcations of Coupled van der Pol–Duffing Oscillators with Multiple Delays

2017 ◽  
Vol 27 (09) ◽  
pp. 1750133 ◽  
Author(s):  
Xia Liu ◽  
Tonghua Zhang
Keyword(s):  
Delay Differential Equations ◽  
Time Delays ◽  
Normal Forms ◽  
Multiple Delays ◽  
Third Order ◽  
Van Der Pol ◽  
Normal Form Theory ◽  
Form Theory ◽  
Delay Differential ◽  
Theoretical Results

In this paper, the Bogdanov–Takens (B–T) and triple zero bifurcations are investigated for coupled van der Pol–Duffing oscillators with [Formula: see text] symmetry, in the presence of time delays due to the intrinsic response and coupling. Different from previous works, third order unfolding normal forms associated with B–T and triple zero bifurcations are needed, which are obtained by using the normal form theory of delay differential equations. Numerical simulations are also presented to illustrate the theoretical results.

scholarly journals An Algorithm for Higher Order Hopf Normal Forms

1995 ◽  
Vol 2 (4) ◽  
pp. 307-319 ◽  
Author(s):  
A.Y.T. Leung ◽  
T. Ge
Keyword(s):  
Normal Form ◽  
Normal Forms ◽  
Nonlinear Differential Equations ◽  
Higher Order ◽  
Qualitative Behavior ◽  
Van Der Pol ◽  
Normal Form Theory ◽  
System A ◽  
Form Theory ◽  
Cubic System

Normal form theory is important for studying the qualitative behavior of nonlinear oscillators. In some cases, higher order normal forms are required to understand the dynamic behavior near an equilibrium or a periodic orbit. However, the computation of high-order normal forms is usually quite complicated. This article provides an explicit formula for the normalization of nonlinear differential equations. The higher order normal form is given explicitly. Illustrative examples include a cubic system, a quadratic system and a Duffing–Van der Pol system. We use exact arithmetic and find that the undamped Duffing equation can be represented by an exact polynomial differential amplitude equation in a finite number of terms.

STEADY-STATE, HOPF AND STEADY-STATE-HOPF BIFURCATIONS IN DELAY DIFFERENTIAL EQUATIONS WITH APPLICATIONS TO A DAMPED HARMONIC OSCILLATOR WITH DELAY FEEDBACK

2012 ◽  
Vol 22 (12) ◽  
pp. 1250286 ◽  
Author(s):  
YONGLI SONG ◽  
JIAO JIANG
Keyword(s):  
Steady State ◽  
Normal Form ◽  
Differential Equations ◽  
Harmonic Oscillator ◽  
Delay Differential Equations ◽  
Hopf Bifurcations ◽  
Normal Form Theory ◽  
Damped Harmonic Oscillator ◽  
Form Theory ◽  
Delay Differential

In this paper, employing the normal form theory of delay differential equations due to Faria and Magalhães, we present explicit formulas of the coefficients of a normal form associated with the flow on a center manifold with the unfolding for general delay differential equations under the cases of steady-state, Hopf and steady-state-Hopf singularities. The explicit conditions determining the transcritical and pitchfork bifurcations for steady-state singularity, determining the direction and stability of Hopf bifurcations, and determining the coefficients of a normal form with universal unfolding for steady-state-Hopf singularity up to third order are obtained. Using the obtained results, we give a complete description of bifurcation scenario of the damped harmonic oscillator with delay feedback near the zero equilibrium. Finally, numerical simulations are given to illustrate our theoretical results and some numerical extensions are obtained as a supplement to our theoretical analysis.

Codimension Bifurcation Analysis of a Modified Leslie–Gower Predator–Prey Model with Two Delays

2018 ◽  
Vol 28 (05) ◽  
pp. 1850060 ◽  
Author(s):  
Jianfeng Jiao ◽  
Ruiqi Wang ◽  
Hongcui Chang ◽  
Xia Liu
Keyword(s):  
Time Delays ◽  
Center Manifold ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Prey Model ◽  
Form Theory ◽  
Two Delays ◽  
Delay Differential

The Bogdanov–Takens (B–T) and triple-zero bifurcations of a modified Leslie–Gower predator–prey model with two time delays are studied in this paper. By generalizing and using the normal form theory and center manifold theorem for delay differential equations, the normal forms of the B–T and triple-zero bifurcations of the model at its interior equilibria are obtained. In addition, some numerical simulations are presented to illustrate our main results.

scholarly journals Stability and Hopf Bifurcation Analysis for a Gause-Type Predator-Prey System with Multiple Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Juan Liu ◽  
Changwei Sun ◽  
Yimin Li
Keyword(s):  
Hopf Bifurcation ◽  
Sufficient Conditions ◽  
Multiple Delays ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Prey System ◽  
Form Theory ◽  
Two Delays ◽  
The Stability ◽  
Theoretical Results

This paper is concerned with a Gause-type predator-prey system with two delays. Firstly, we study the stability and the existence of Hopf bifurcation at the coexistence equilibrium by analyzing the distribution of the roots of the associated characteristic equation. A group of sufficient conditions for the existence of Hopf bifurcation is obtained. Secondly, an explicit formula for determining the stability and the direction of periodic solutions that bifurcate from Hopf bifurcation is derived by using the normal form theory and center manifold argument. Finally, some numerical simulations are carried out to illustrate the main theoretical results.

Delay-Induced Double Hopf Bifurcations in a System of Two Delay-Coupled van der Pol–Duffing Oscillators

2015 ◽  
Vol 25 (04) ◽  
pp. 1550058 ◽  
Author(s):  
Heping Jiang ◽  
Tonghua Zhang ◽  
Yongli Song
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Hopf Bifurcations ◽  
Van Der Pol ◽  
Normal Form Theory ◽  
Double Hopf Bifurcation ◽  
Codimension Two ◽  
Form Theory ◽  
Weak Resonance ◽  
Delay Differential

In this paper, we investigate the codimension-two double Hopf bifurcation in delay-coupled van der Pol–Duffing oscillators. By using normal form theory of delay differential equations, the normal form associated with the codimension-two double Hopf bifurcation is calculated. Choosing appropriate values of the coupling strength and the delay can result in nonresonance and weak resonance double Hopf bifurcations. The dynamical classification near these bifurcation points can be explicitly determined by the corresponding normal form. Periodic, quasi-periodic solutions and torus are found near the bifurcation point. The numerical simulations are employed to support the theoretical results.

scholarly journals Equivariant Hopf Bifurcation in a Ring of Identical Cells with Delay

2009 ◽  
Vol 2009 ◽  
pp. 1-34 ◽  
Author(s):  
Dejun Fan ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Functional Differential Equations ◽  
Hopf Bifurcations ◽  
Normal Form Theory ◽  
Form Theory ◽  
Functional Differential ◽  
Delay Differential ◽  
Equivariant Hopf Bifurcation

A kind of delay neural network withnelements is considered. By analyzing the distribution of the eigenvalues, a bifurcation set is given in an appropriate parameter space. Then by using the theory of equivariant Hopf bifurcations of ordinary differential equations due to Golubitsky et al. (1988) and delay differential equations due to Wu (1998), and combining the normal form theory of functional differential equations due to Faria and Magalhaes (1995), the equivariant Hopf bifurcation is completely analyzed.

Analysis of Oscillatory Patterns of a Discrete-Time Rosenzweig–MacArthur Model

2018 ◽  
Vol 28 (06) ◽  
pp. 1850075
Author(s):  
Xiaoqin P. Wu ◽  
Liancheng Wang
Keyword(s):  
Discrete Time ◽  
Normal Forms ◽  
Critical Values ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Positive Equilibrium Point ◽  
Associated Species ◽  
Theoretical Results

This article investigates the oscillatory patterns of the following discrete-time Rosenzweig–MacArthur model [Formula: see text] The system describes the evolution and interaction of the populations of two associated species (prey and predator) from generation to generation. We show that this system can exhibit co-dimension-1 bifurcations (flip and Neimark–Sacker bifurcations) as [Formula: see text] crosses some critical values and codimension-2 bifurcations (1:2, 1:3, and 1:4 resonances) for certain critical values of [Formula: see text] at the positive equilibrium point. The normal form theory and the center manifold theorem are used to obtain the normal forms. For codimension-2 bifurcations, the bifurcation diagrams are established by using these normal forms along the orbits of differential equations. Numerical simulations are presented to confirm the theoretical results.

scholarly journals Dynamical Behavior in a Four-Dimensional Neural Network Model with Delay

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Changjin Xu ◽  
Peiluan Li
Keyword(s):  
Neural Network ◽  
Hopf Bifurcation ◽  
Network Model ◽  
Neural Network Model ◽  
Center Manifold ◽  
Dynamical Behavior ◽  
Normal Form Theory ◽  
Form Theory ◽  
Explicit Formulae ◽  
Delay Differential

A four-dimensional neural network model with delay is investigated. With the help of the theory of delay differential equation and Hopf bifurcation, the conditions of the equilibrium undergoing Hopf bifurcation are worked out by choosing the delay as parameter. Applying the normal form theory and the center manifold argument, we derive the explicit formulae for determining the properties of the bifurcating periodic solutions. Numerical simulations are performed to illustrate the analytical results.

scholarly journals Stability and bifurcation in a simplified four-neuron BAM neural network with multiple delays

2006 ◽  
Vol 2006 ◽  
pp. 1-29 ◽  
Author(s):  
Xiang-Ping Yan ◽  
Wan-Tong Li
Keyword(s):  
Neural Network ◽  
Critical Values ◽  
Multiple Delays ◽  
Multiple Time ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Fourth Degree ◽  
Form Theory ◽  
Bam Neural Network ◽  
The Stability

We first study the distribution of the zeros of a fourth-degree exponential polynomial. Then we apply the obtained results to a simplified bidirectional associated memory (BAM) neural network with four neurons and multiple time delays. By taking the sum of the delays as the bifurcation parameter, it is shown that under certain assumptions the steady state is absolutely stable. Under another set of conditions, there are some critical values of the delay, when the delay crosses these critical values, the Hopf bifurcation occurs. Furthermore, some explicit formulae determining the stability and the direction of periodic solutions bifurcating from Hopf bifurcations are obtained by applying the normal form theory and center manifold reduction. Numerical simulations supporting the theoretical analysis are also included.

scholarly journals The Time Delays’ Effects on the Qualitative Behavior of an Economic Growth Model

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Carlo Bianca ◽  
Massimiliano Ferrara ◽  
Luca Guerrini
Keyword(s):  
Economic Growth ◽  
Growth Model ◽  
Time Delays ◽  
Sufficient Conditions ◽  
Qualitative Behavior ◽  
Solow Model ◽  
Normal Form Theory ◽  
Economic Growth Model ◽  
Form Theory ◽  
The Stability

A further generalization of an economic growth model is the main topic of this paper. The paper specifically analyzes the effects on the asymptotic dynamics of the Solow model when two time delays are inserted: the time employed in order that the capital is used for production and the necessary time so that the capital is depreciated. The existence of a unique nontrivial positive steady state of the generalized model is proved and sufficient conditions for the asymptotic stability are established. Moreover, the existence of a Hopf bifurcation is proved and, by using the normal form theory and center manifold argument, the explicit formulas which determine the stability, direction, and period of bifurcating periodic solutions are obtained. Finally, numerical simulations are performed for supporting the analytical results.

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